Origins of Mathematical/Logical questions of computability...

[Scott Mayers]

I'm opening this thread in light of the many threads of interest here on computation, logic, and math. This may act as a reference for everyone to discuss the varying histories about the topic. I'm not interested in our differences of opinion but to express each person's understandings of any various interesting histories ABOUT the subject or related to.

Pythagoreans aim to relate every number as 'ratios'
I'll begin my own input by going back in time to the Pythagorean interest in making sense of number by 'comparing' finite whole numbers to other finite whole numbers.

The concept of number was 'experimented' on by taking natural numbered concepts as an original assumption and then relate them to visual measures geometrically. As such, the idea was to begin with DISCRETE counts (natural positive numbers) in visual geometric forms in an attempt to 'rationalize' CONTINUOUS numbers that reality seems to be about.

For instance, if given some unit measure, say a linear segment, can you progressively use a minimal of mechanical procedures to discover all truths about numbers, and thus, by extension to interpreting reality as 'geometric', to link the logic of numbers to reality.

If given a unit segment, you can mechanically present a segment twice its length using only a protractor and a straight-edge ruler. If what is true about doubling some unit measure in a mechanical way, it is conceivable to also half the unit to discover parts of these wholes. Euclid summarized much of what came before his in his "Elements" in a formalized process beginning in the fewest assumed postulates (or axioms).

In their day, what was 'rational' was what could be demonstrated in terms of FINITE comparisons. What we say is one half is just a 'ratio' of two whole numbers, such as 1:2. That is, they only defined the continuous points between two endpoints of a segment as some comparison of two natural numbers. What they didn't expect was to discover that even if they could finitely express all points as natural-number relationships, some real points in between any two rational numbers still existed. This presented a crisis for them because it didn't make sense that you could have something 'irrational' existing.

This is the first "incompleteness" concept. That using natural numbered units and fractions using ratios, you could not COMPLETELY cover all real problems. This hurt the Pythagorean confidence that logic (via math) was able to prove that all of reality was based merely on the concept of 'number'.

The proof of this came about by showing that the square root of 2 could not be presented as a ratio of two natural numbers. The reality of the Pythagoras Theorem said that if we have any two perpendicular segments joined at a vertices of a triangle, the third side is some number such that the square of the two legs of this right triangle is equal to the square on the third, its hypotenuse.

Here is the summary proof :

Irrationality of square-root of 2.png

I arbitrarily picked this from Proof by Contradiction from the "Art of Problem Solving" site.

This shows that what they assumed was "rational" was not actually the 'complete' truth about reality. THIS is one of the earliest 'incompleteness' proofs which sparks concern about the limitations of logic. However, note that this 'problem' was later resolved when we learned to understand irrational numbers 'rationally' (a contradiction in old terms).

[Speakpigeon]

Alright, I'm not going to argue with you but there's an aspect of the history of logic that interests me. As I see it, people who started logic and maths, essentially Aristotle and Euclid, and indeed before them Pythagoras, sort of, seem to have been interested mainly in the usefulness and applicability of their concepts. They also apparently started from the empirical evidence available to them. However, after Aristotle, the Stoics already seem to consider logic in the abstract, independently of the actual truth of the premises considered, unlike Aristotle. I'm not sure of the intervening history but the Scholastics were queen on taking logic in the abstract, too. Essentially, I think, because they were biased in this direction. Their interest was entirely, officially at least, in arguing among themselves about God, a topic where actual truths are not generally available. When science started to develop after Copernicus had given the signal, logic, as an abstract and so-called "a priori" discipline, got trashed in favour of a posteriori knowledge based on empirical evidence, as the linchpin of science. Descartes may have been the one to have put the knife in. His views I think were nonsense but he seemed to have been very influential. Then people like Hume and Locke. Meanwhile, mathematics was getting more abstract, starting with Newton's and Leibniz's development of Calculus and consideration of the infinite. It soon became received wisdom among many mathematicians that mathematics was essentially independent of empirical truths. I guess Cantor's work on the infinite is the epitome of this view.

OK, I'll stop here for now. It's a complex question, and there are several aspects to it. In particular, I see an air de déjà vu in the criticisms levelled at the conventional view of mathematics coming broadly from Turing-machine inspired Constructivism, as it reminds me of the criticisms of logic by empiricists following Descartes.

Of course, logic didn't disappear, as far as I know.
EB